1. Field of the Invention
The present invention relates to a multi-scale analysis device for performing structural analysis on a multi-scale, from global (macro) structures to micro structures, using a finite element method.
2. Description of the Related Art
It is realistically impossible to analyze entire structures on a minute scale when analyzing large-scale structures such as ships, airplanes and automobiles, or structures, such as electronic devices, having a multi-layer structure from entire meter-order devices to nanometer-order LSI (large scale integration) internal structures. This is because, in such minute analyses, computer capacity may be insufficient or modeling may require enormous costs.
However, in order to evaluate the fatigue strength of a fine structure part and the like, it is necessary to determine the stress concentration due to its minute shape, and modeling of the minute shape is essential. Therefore, in the analysis of structures such as these, a method commonly called zooming analysis is used.
With this method, first, the entire behavior is analyzed, providing load conditions such as load and temperatures to a global model created with rough finite element mesh by simplifying the entire structure. Then, a model wherein minute mesh division is performed to the area of which minute stress is to be determined is created separately, appropriate boundary conditions are provided based on the results of the global analysis, and minute analysis is performed. Generally, in electronic devices, the structure is divided into a number of levels, such as the entire device, unit parts, LSI package, LSI internal wiring, and zooming analysis is performed in multi-levels.
FIG. 1A shows an example of a zooming analysis of an electronic device such as this. In this example, package 811 is fixed to substrate 813 by soldering 812, and a fine mesh micro-model 802 is formed on a part of a rough mesh global model 801.
A displacement or load which is obtained by an analysis in one-level-higher scale is given as the boundary condition in the zooming analysis. In many cases, a micro analysis is performed with a displacement obtained by a global analysis as a displacement in a micro-structure.
However, in zooming analysis, because minute analysis is performed using the results of global analysis by mesh which is rougher in comparison to a micro-structure, accuracy is an issue. When creating models for preventing this issue, large amounts of experience are required in the modeling method, such as appropriate mesh division in global structures and micro-structures, definition of boundary nodes in micro-structures and the like.
As conventional methods for determining the node displacement of a micro-structure, the following two methods are given:                (1) Matching the position of the boundary node of the micro-structure to the node position of the global structure        (2) Determining the boundary displacement of the micro-structure by interpolating the displacement of the global analysis result.        
In the foregoing method (1), as shown in FIG. 1B, the coordinate position of the boundary node of the micro model 902 completely matches the node position of the global model 901. In this case, the mesh of global model 901 and the mesh of micro model 902 are appropriately divided as boundary conditions, and the node which provides the boundary conditions of micro model 902 (shown as ∘ in FIG. 1B) must completely match the node of the mesh of the global model 901.
Here, if the boundary node displacement of the micro-structure is um and the analysis result displacement of the global structure is ug, the following expression can be obtained:um(x,y,z)=ug(x,y,z)  (1)
In expression (1), x, y and z indicate space coordinates, and expression (1) indicates defining of the boundary displacement of a micro-structure using the node displacement of the global analysis result with regards to nodes with equal space coordinates in global structure and micro structure.
When performing zooming analysis, specifically, by finite element method, either one of the following two methods is used.    a) The operator reads the node displacement of a global structure from the results of a global analysis and manually adds this to the boundary node of a micro-structure as boundary conditions. Although this method is effective because it can be realized without depending on a simulation program when the number of boundary nodes of the micro-structure is small, it is not realistic when the number of boundary nodes increase.    b) A boundary node group in a micro analysis is defined first, and in the preparation stage prior to the execution of micro-structure simulation, global structure nodes with coordinates which match the space coordinates of the nodes belonging to the boundary node group of the micro-structure are retrieved from the results of global analysis. Then, a micro analysis is performed using displacements of these nodes. In this method, the simulation program per se must have a zooming analysis function or a program for performing zooming processing must be prepared separately.
In the foregoing method (2), the boundary node displacement of the micro-structure is determined by interpolating the node displacement of the global analysis results. In this method, as shown in FIG. 1B, it is not necessary for the node coordinates of micro model 903 to match the node coordinate positions of global model 901, and therefore, in comparison with method (1), the mesh of micro model 903 can be created freely.
In this case, although it is necessary to interpolate node displacement providing boundary conditions of the micro-structure from the global analysis results, various methods can be considered. Below, a two-dimensional plane problem is assumed and, as shown in FIG. 1C, a method for performing interpolation by considering the three global nodes in the vicinity of the micro model node to be a triangular element and using a shape function of the triangular element is described.
In FIG. 1C, nodes i, j, and k are nodes on the mesh of a global structure and node M is a node on the mesh of a micro structure. If the space coordinates of the node is (x, y), and displacements in the x direction and the y direction are respectively u and v, the coordinates and displacements of the four nodes are indicated as follows:    i coordinates (xi, yi) displacements (ui, vi)    j coordinates (xj, yj) displacements (uj, vj)    k coordinates (xk, yk) displacements (uk, vk)    M coordinates (xm, ym) displacements (um, vm)
Inside the triangular element, displacement is approximated as a linear function with respect to space coordinates.u(x,y)=a+bx+cy v(x,y)=d+ex+fy  (2)
Node displacements (ui, vi) to (uk, vk) must also fulfill expression (2), and therefore, the following expressions are obtained.
                                          (                                                            1                                                  xi                                                  yi                                                                              1                                                  xj                                                  yj                                                                              1                                                  xk                                                  yk                                                      )                    ⁢                      (                                                            a                                                                              b                                                                              c                                                      )                          =                  (                                                    ui                                                                    uj                                                                    uk                                              )                                    (        3        )                                                      (                                                            1                                                  xi                                                  yi                                                                              1                                                  xj                                                  yj                                                                              1                                                  xk                                                  yk                                                      )                    ⁢                      (                                                            d                                                                              e                                                                              f                                                      )                          =                  (                                                    vi                                                                    vj                                                                    vk                                              )                                    (        4        )            
Because (xi, yi) to (xk, yk) in expression (3) and expression (4) are coordinates of global nodes, it is known at the time global analysis is completed. Therefore, if ui to uk and vi to vk are known, respective in variables, a to f, can be determined from the inverse matrices of expression (3) and expression (4). For example, the inverse matrix of expression (3) is as follows:
                    (                                                                                                  -                    xkyj                                    +                  xjyk                                Δ                                                                                      xkyi                  -                  xiyk                                Δ                                                                                                          -                    xjyi                                    +                  xiyj                                Δ                                                                                                          yj                  -                  yk                                Δ                                                                                                          -                    yi                                    +                  yk                                Δ                                                                                      yi                  -                  yj                                Δ                                                                                                                              -                    xj                                    +                  xk                                Δ                                                                                      xi                  -                  xk                                Δ                                                                                                          -                    xi                                    +                  xj                                Δ                                                    )                            (        5        )            
Denominator Δ common to respective matrix elements is expressed as follows:Δ=−xjyi+xkyi+xiyj−xkyj−xiyk+xjyk  (6)
If the inverse matrix of (5) is multiplied from the left side of expression (3), the following expressions can be obtained as expressions to determine invariables a, b, and c.a=[uk(−xjyi+xiyj)+uj(xkyi−xiyk)+ui(−xkyj+xjyk)]/Δb=[uk(yi−yj)+ui(yj−yk)+uj(yk−yj)]/Δc=[uk(xi−xj)+ui(xj−xk)+uj(xk−xj)]/Δ  (7)
In the same way, if the inverse matrix of (5) is multiplied from the left side of expression (4), the following expressions, wherein u in expression (7) are replaced with v, can be obtained as expressions to determine invariables d, e, and f.d=[vk(−xjyi+xiyj)+vj(xkyi−xiyk)+vi(−xkyj+xjyk))]/Δe=[vk(yi−yj)+vi(yj−yk)+vj(yk−yj)]/Δf=[vk(xi−xj)+vi(xj−xk)+vj(xk−xj)]/Δ  (8)
The boundary node displacements of node M of the micro-structure, (um, vm), can be calculated as follows using the invariables determined in expression (7) and expression (8).um(xm,ym)=a+bxm+cym vm(xm,ym)=d+exm+fym  (9)
Although two-dimensional problems are used as examples in the foregoing calculations, in instances of one-dimensional problems, it is a simple linear interpolation. In addition, if it is a three-dimensional problem, a tetrahedral element composed of neighboring four nodes of a global model is assumed and calculations similar to those for the two-dimensional problems are performed.
The patent references 1 to 3, below, are related to a structure analysis using a finite element method.    Patent Reference 1: Japan Patent Application Publication No. 2001-350802    Patent Reference 2: Japan Patent Application Publication No. 2001-027994    Patent Reference 3: Japan Patent Application Publication No. 10-334276
Out of the conventional methods described above for determining the node displacements of a micro-structure, in the method for determining the node displacements of the micro-structure from the displacements of global analysis results through the linear interpolation in expression (9), if the average mesh size of the global model is Δx, an approximation error proportional to Δx2 is generated. Therefore, there are limits with regards to accuracy, and in order to obtain accurate analysis results, Δx must be set to a small value and the computer processing time required for global analysis increases.